Friday, January 17, 2020

Hillcrest applies quadratic equation to bottle rockets

Hillcrest H.S. algebra 1 students launch bottle rockets (school newspaper)

Algebra 1 students at Hillcrest High School in Country Club Hills, Illinois, took to the school lawn this spring to launch bottle rockets and apply the quadratic equation in the process, The Hawk Eye reports. It’s an annual warm-weather tradition at the south-suburban school for students of math teacher Kara Bucci, says the student newspaper.

Height, 8-second bottle rocket flight

The vertical height, or altitude, of a bottle rocket can be calculated or, at least, approximated by knowing that the height plotted against time is a parabola, assuming no friction and a straight-up flight. The equation for the maximum height achieved by a bottle rocket that is launched from Earth, where friction is an actual component, is given by a formula known as Littlewood’s law, named after JE Littlewood, a mathematician in the British military during World War I.

Although this isn’t a “law” in the scientific sense, Mr Littlewood’s equation relates the altitude at apogee to the total duration of the flight, from the time the rocket is launched to the time it returns to the ground:

$h=0.122 \cdot g \cdot t^2$

where h is the altitude at apogee, t is the total duration of the flight from launch to touchdown, and g is the acceleration of gravity, which is about 9.8 m/s2.

Say the rocket stays in the air for exactly 8 seconds, as determined by a student with a stopwatch. We can calculate the approximate height at apogee as follows:

$h=(0.122 \cdot 9.8 \cdot 8^2)\textrm{ m}$
$h \approx 76.52\textrm{ m} \approx 250\textrm{ feet}$
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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